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Determine the angle $$x$$

Solution

Starting by adding some lengths like it is shown in the next figure
Now, let’s take the triangles in the two next figures
We have $\tan20°=\frac{d}{a}$ $and$ $\tan50°=\frac{d}{b}$ $\Rightarrow d=b\tan50°=a\tan20°$ $\Rightarrow\frac{a}{b}=\frac{\tan50°}{\tan20°}$
For the next triangles
$\tan10°=\frac{c}{a}$
$\tan x=\frac{c}{b}$
$\Rightarrow c=b\tan x=a\tan10°$ $\Rightarrow\frac{a}{b}=\frac{\tan x}{\tan10°}$
Using the two results above, we get
$\frac{\tan x}{\tan10°}=\frac{\tan50°}{\tan20°}$ $\Rightarrow\tan x=\frac{\tan50°}{\tan20°}\left(\tan10°\right)$
$\Rightarrow x=\arctan\left(\frac{\tan50°}{\tan20°}\left(\tan10°\right)\right)=30°$
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